Skip to main content
SpringerLink
Log in

Disjointness of the Möbius Transformation and Möbius Function

  • Research
  • Published:
Research in the Mathematical Sciences Aims and scope Submit manuscript

Abstract

We study the distribution of the sequence of elements of the discrete dynamical system generated by the Möbius transformation \(x \mapsto (ax + b)/(cx + d)\) over a finite field of p elements. Motivated by a recent conjecture of P. Sarnak, we obtain nontrivial estimates of exponential sums with such sequences that imply that trajectories of this dynamical system are disjoined with the Möbius function.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baake, M., Neumärker, N., Roberts, J.A.G.: Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices. Discrete Cont. Dyn. Syst. Ser. A 33, 527–553 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Banks, W.D., Conflitti, A., Friedlander, J.B., Shparlinski, I.E.: Exponential sums over Mersenne numbers. Compos. Math. 140, 15–30 (2004)

    Article  MathSciNet  Google Scholar 

  3. Banks, W.D., Friedlander, J.B., Garaev, M.Z., Shparlinski, I.E.: Double character sums over elliptic curves and finite fields. Pure Appl. Math. Q. 2, 179–197 (2006)

    Article  MathSciNet  Google Scholar 

  4. Banks, W.D., Friedlander, J.B., Garaev, M.Z., Shparlinski, I.E.: Exponential and character sums Mersenne numbers. J. Aust. Math. Soc. 92, 1–13 (2012)

    Article  MathSciNet  Google Scholar 

  5. Blümel, R., Reinhardt, W.P.: Chaos in Atomic Physics. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  6. Bourgain, J.: On the maximal ergodic theorem for certain subsets of the integers. Isr. J. Math. 61, 39–72 (1988)

    Article  MathSciNet  Google Scholar 

  7. Bourgain, J.: On the pointwise ergodic theorem on \(L^p\) for arithmetic sets. Isr. J. Math. 61, 73–84 (1988)

    Article  Google Scholar 

  8. Bourgain, J.: An approach to pointwise ergodic theorems. In: Geometric Aspects of Functional Analysis (1986/87), Lecture Notes in Math., vol. 1317, pp. 204–223. Springer, Berlin (1988)

  9. Bourgain, J.: Pointwise ergodic theorems for arithmetic sets (with an appendix by J. Bourgain, H. Furstenberg, Y Katznelson and D.S. Ornstein). Inst. Hautes Études Sci. Publ. Math. 69, 5–45 (1989)

    Article  Google Scholar 

  10. Bourgain, J.: Estimates on exponential sums related to Diffie-Hellman distributions. Geom. Funct. Anal. 15, 1–34 (2005)

    Article  MathSciNet  Google Scholar 

  11. Bourgain, J.: A remark on quantum ergodicity for CAT maps. In: Geometric Aspects of Functional Analysis, Lecture Notes in Math., vol. 1910, pp. 89–98. Springer, Berlin (2007)

  12. Bourgain, J.: On the correlation of the Möbius function with rank-one systems. J. Anal. Math. 120, 105–130 (2013)

    Article  MathSciNet  Google Scholar 

  13. Bourgain, J., Sarnak, P., Ziegler, T.: Disjointness of Möbius from horocycle flow. In: From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Devel. Math., vol. 28, pp. 67–83. Springer, New York (2013)

  14. Buczolich, Z.: Ergodic averages with prime divisor weights in \(L^1\). Ergod. Theory Dyn. Syst. (2017). https://doi.org/10.1017/etds.2017.54

  15. Carmon, D., Rudnick, Z.: The autocorrelation of the Möbius function and Chowla’s conjecture for the rational function field. Q. J. Math. 65, 53–61 (2014)

    Article  MathSciNet  Google Scholar 

  16. Chou, W.-S.: On inversive maximal period polynomials over finite fields. Appl. Algebra Eng. Commun. Comput. 6, 245–250 (1995)

    Article  MathSciNet  Google Scholar 

  17. Cuny, C., Weber, M.: Ergodic theorems with arithmetical weights. Isr. J. Math. 217, 139–180 (2017)

    Article  MathSciNet  Google Scholar 

  18. Daboussi, H., Delange, H.: On multiplicative arithmetical functions whose modulus does not exceed one. J. Lond. Math. Soc. 26, 245–264 (1982)

    Article  MathSciNet  Google Scholar 

  19. el Abdalaoui, E.H.: On Veech’s proof of Sarnak’s theorem on the Möbius flow. Preprint (2017). arXiv:1711.06326

  20. el Abdalaoui, E.H., Lemańczyk, M., de la Rue, T.: On spectral disjointness of powers for rank-one transformations and Möbius orthogonality. J. Funct. Anal. 266, 284–317 (2014)

    Article  MathSciNet  Google Scholar 

  21. Eisner, T.: A polynomial version of Sarnak’s conjecture. C. R. Math. Acad. Sci. Paris 353, 569–572 (2015)

    Article  MathSciNet  Google Scholar 

  22. el Abdalaoui, E.H., Kułaga-Przymus, J., Lemańczyk, M., de la Rue, T.: The Chowla and the Sarnak conjectures from ergodic theory point of view. Discrete Cont. Dyn. Syst. Ser. A 37, 2899–2944 (2017)

    Article  MathSciNet  Google Scholar 

  23. Eisner, T., Lin, M.: On modulated ergodic theorems. Preprint (2017). arXiv:1709.05322

  24. Everest, G., van der Poorten, A., Shparlinski, I.E., Ward, T.: Recurrence sequences. In: Math. Surveys and Monogr., vol. 104. Amer. Math. Soc., Providence (2003)

  25. Ferenczi, S., Kułaga-Przymus, J., Lemańczyk, M.: Sarnak’s conjecture—what’s new. In: Ergodic Theory and Dynamical Systems in Their Interactions with Arithmetics and Combinatorics, Lecture Notes in Mathematics, vol. 2213, pp. 163–235. Springer, Berlin (2018)

  26. Fouvry, É., Ganguly, S.: Strong orthogonality between the Möbius function, additive characters and Fourier coefficients of cusp forms. Compos. Math. 150, 763–797 (2014)

    Article  MathSciNet  Google Scholar 

  27. Friedlander, J.B., Hansen, J., Shparlinski, I.E.: On character sums with exponential functions. Mathematika 47, 75–85 (2000)

    Article  MathSciNet  Google Scholar 

  28. Garaev, M.Z., Shparlinski, I.E.: The large sieve inequality with exponential functions and the distribution of Mersenne numbers modulo primes. Int. Math. Res. Not. 39, 2391–2408 (2005)

    Article  MathSciNet  Google Scholar 

  29. Gomilko, A., Kwietniak, D., Lemańczyk, M.: Sarnak’s conjecture implies the Chowla conjecture along a subsequence. In: Ergodic Theory and Dynamical Systems in Their Interactions with Arithmetics and Combinatorics, Lecture Notes in Math., vol. 2213, pp. 237–247. Springer, Cham (2018). arXiv:1710.07049

  30. Green, B.J.: A note on multiplicative functions on progressions to large moduli. Proc. R. Soc. Edinb. 148A, 63–77 (2018)

    Article  MathSciNet  Google Scholar 

  31. Green, B.J., Tao, T.: The Möbius function is strongly orthogonal to nilsequences. Ann. Math. 175, 541–566 (2012)

    Article  MathSciNet  Google Scholar 

  32. Iwaniec, H., Kowalski, E.: Analytic Number Theory. Amer. Math. Soc., Providence (2004)

    MATH  Google Scholar 

  33. Kátai, I.: A remark on a theorem of H. Daboussi. Acta Math. Hungar. 47, 223–225 (1986)

    Article  MathSciNet  Google Scholar 

  34. Kelmer, D.: On matrix elements for the quantized cat cap modulo prime powers. Ann. Henri Poincaré 9, 1479–1501 (2008)

    Article  MathSciNet  Google Scholar 

  35. Kułaga-Przymus, J., Lemańczyk, M.: The Möbius function and continuous extensions of rotations. Monat. Math. 178, 553–582 (2015)

    Article  Google Scholar 

  36. Kurlberg, P.: Bounds on supremum norms for Hecke eigenfunctions. Ann. Henri Poincaré 8, 75–89 (2007)

    Article  MathSciNet  Google Scholar 

  37. Kurlberg, P., Rosenzweig, L., Rudnick, Z.: Matrix elements for the quantum cat map: fluctuations in short windows. Nonlinearity 20, 2289–2304 (2007)

    Article  MathSciNet  Google Scholar 

  38. Kurlberg, P., Rudnick, Z.: Hecke theory and equidistribution for the quantization of linear maps of the torus. Duke Math. J. 103, 47–77 (2000)

    Article  MathSciNet  Google Scholar 

  39. Kurlberg, P., Rudnick, Z.: On the distribution of matrix elements for the quantum cat map. Ann. Math. 161, 489–507 (2005)

    Article  MathSciNet  Google Scholar 

  40. Li, W.-C.W.: Number Theory with Applications. World Scientific, Singapore (1996)

    Book  Google Scholar 

  41. Li, W.-C.W.: Character sums over \(p\)-adic fields. J. Number Theory 174, 181–229 (1999)

    Article  MathSciNet  Google Scholar 

  42. Li, W.-C.W.: Character sums over norm groups. Finite Fields Appl. 12, 1–15 (2006)

    Article  MathSciNet  Google Scholar 

  43. Liu, J., Sarnak, P.: The Möbius disjointness conjecture for distal flows. Duke Math. J. 164, 1353–1399 (2015)

    Article  MathSciNet  Google Scholar 

  44. Marchetti, D.H.U., Wreszinski, W.F.: Asymptotic Time Decay in Quantum Physics. World Scientific, Singapore (2013)

    Book  Google Scholar 

  45. Nair, R.: On polynomials in primes and J. Bourgain’s circle method approach to ergodic theorems. Ergod. Theory Dyn. Syst. 11, 485–499 (1991)

    Article  MathSciNet  Google Scholar 

  46. Nair, R.: On polynomials in primes and J. Bourgain’s circle method approach to ergodic theorems II. Studia Math. 105, 207–233 (1993)

    Article  MathSciNet  Google Scholar 

  47. Ostafe, A., Shparlinski, I.E.: Exponential sums over points of elliptic curves with reciprocals of primes. Mathematika 58, 21–33 (2012)

    Article  MathSciNet  Google Scholar 

  48. Ostafe, A., Shparlinski, I.E.: On the power generator of pseudorandom numbers and its multivariate analogue. J. Complexity 28, 238–249 (2012)

    Article  MathSciNet  Google Scholar 

  49. Ramaré, O.: Arithmetical aspects of the large sieve inequality (with the collaboration of D. S. Ramana). In: Harish–Chandra Res. Inst. Lecture Notes, vol. 1. Hindustan Book Agency, New Delhi (2009)

  50. Rosenblatt, J.M., Wierdl, M.: Pointwise ergodic theorems via harmonic analysis. In: Ergodic Theory and Its Connections with Harmonic Analysis (Alexandria, 1993), London Math. Soc. Lecture Note Ser., vol. 205, pp. 3–151. Cambridge University Press, Cambridge (1995)

  51. Rosenzweig, L.: Fluctuations of matrix elements of the quantum cat map. Int. Math. Res. Not. 2011, 4884–4933 (2011)

    MathSciNet  MATH  Google Scholar 

  52. Ryzhikov, V.V.: Bounded ergodic constructions, disjointness, and weak limits of powers. Trans. Moscow Math. Soc. 74, 165–171 (2013)

    Article  MathSciNet  Google Scholar 

  53. Sarnak, P.: Möbius randomness and dynamics. Not. S. Afr. Math. Soc. 43, 89–97 (2012)

    MathSciNet  Google Scholar 

  54. Sarnak, P., Ubis, A.: The horocycle flow at prime times. J. Math. Pures Appl. 103, 575–618 (2015)

    Article  MathSciNet  Google Scholar 

  55. Tao, T.: Equivalence of the logarithmically averaged Chowla and Sarnak conjectures. In: Elsholtz, C., Grabner, P. (eds.) Number Theory—Diophantine Problems, Uniform Distribution and Applications; Festschrift in Honour of Robert F. Tichy’s 60th Birthday, pp. 391–421. Springer, Berlin (2017)

  56. Tao, T., Teräväinen, J.: The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures. Duke Math. J. (2017)

  57. Thouvenot, J.-P.: La convergence presque sûre des moyennes ergodiques suivant certaines sous-suites d’entiers (d’après Jean Bourgain), Séminaire Bourbaki, vol. 1989/90. Astérisque 189–190, Exp. No. 719, pp. 133–153 (1990)

  58. Weil, A.: Basic Number Theory. Springer, New York (1974)

    Book  Google Scholar 

  59. Wierdl, M.: Pointwise ergodic theorem along the prime numbers. Isr. J. Math. 64, 315–336 (1988)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques Raphaël Salem, Université de Rouen Normandie, 76801, Saint-Étienne-du-Rouvray, France

    El Houcein El Abdalaoui

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    Igor E. Shparlinski

Authors
  1. El Houcein El Abdalaoui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Igor E. Shparlinski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Igor E. Shparlinski.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

El Abdalaoui, E.H., Shparlinski, I.E. Disjointness of the Möbius Transformation and Möbius Function. Res Math Sci 6, 17 (2019). https://doi.org/10.1007/s40687-019-0180-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40687-019-0180-6

Keywords

Mathematics Subject Classification

Search

Navigation